Friday, June 1, 2012

Limit Notation in Calculus

This is going to be a very short post about a crucial subject in calculus: limts!  In my previous post, I gave a short and quite basic explanation of what is meant when we talk about limits in calculus.  I explained that they are similar to a concept that you know in everyday life, such as a speed limit, though they have their differences.  I also simply demonstrated that the value at a point on a curve does not necessarily equal the limit as you approach the point.  They require a different way to think about curves and graphs, in that where you are traditionally used to thinking about what happens exactly at a specific point on a graph (such as, what is the value of y when x equals 2), limits require you to consider what is happening to the curve as you approach a point and get infinitely closer to it.  Essentially, it may be thought of as "based on the shape of the curve that I know for sure, what does it look like the curve is going to do next?"  What it looks like it's going to do, the limit, might not always be exactly what it does do, which would be what the equation of the line would describe exactly.

So, that is a very brief overview of what my last post discussed, and over the next several posts, I will be expanding on this concept of limits, and explaining several different points about them and how to work with them.  First, however, I need to explain the notation for limits.  Limit notation in calculus is quite intuitive when you look at it.  That is, it is quite easy to understand when you see it.  Here is how you would write the symbol for a limit:

To read this, one would read it as "the limit of f of x, as x approaches a, equals L."

The "lim" part indicates that this is a limit, the "f(x)" (in this case) could be any function or expression, and the x→a part indicates the value "a" that x is approaching for which we want to evaluate the limit, L.  Let's consider the following graph again, the same one I drew in my introduction to limits post.

Now, say that we want to determine what the limit of this curve is (let's just call it the generic f(x)) as x→p (*note* assume for now that the point P is designated by the ordered pair (p,y)... tonight I'm too lazy to go back and edit my picture!).  We would write this expression as:

This expression reads as "the limit of f of x, as x approaches p."  Now, in this case, just by looking at the graph, we can visually see that if we trace the curve and move towards point P, where x=p, we can see that the curve is approaching y=4.  In other words, the limit is 4.

Similarly, refer back to my calculus limits introduction post, and you can see on the second example, where y=10 when x=p, the limit as x approaches p is also 4.

I hope that this series of posts is beginning to help you to grasp the concept of limits in calculus.  In my next post, I would like to develop this one a bit further, by introducing the concept of directional limits.  For the examples I've shown so far, we have only considered functions where the curve approaches the same point from both sides.  However, some graphs (e.g. piecewise equations) can have a different limit value, depending on which direction you approach "a" from.  I will explain these one-sided limits and their limit notation in the next post.

Wednesday, May 30, 2012

What are Limits in Calculus?

Back when I was taking my introductory calculus courses, one of the first topics that we covered was a section on limits and rates of change.  To a math student, "rate of change" kind of makes some sense.  It sounds like a math concept that explains how fast something is changing.  It's a fairly intuitive concept, and we will eventually see that it is a very important concept as well!  On the other hand, at first glance, the concept of a "limit" sounds a bit more abstract.  We all know about limits in terms of, for example, the speed limit for a car or other defined measures.  But how does this construct fit into calculus, or does "limit" mean something else entirely in this context?

The concept of a limit is essentially the same as when we're talking about a speed limit.  Let's say there is a posted speed limit of 50 mph.  Your car is not allowed to travel any faster than this.  So, imagine then what a graph would look like of the speed of your car over time.  At time = 0, you start stationary.  Then you accelerate and start driving a bit faster, until you get to the speed limit, at which point you don't go any faster (because we obey the laws!), and then you approach a stop sign and slow down to come to a stop.  When you look at the graph of what has just happened, you can see that the speed limit is basically a speed value that your car approaches.

In calculus, the concept of a limit is very similar, yet there is a slight difference that is very important.  In quite basic terms, a limit of a graph (or more accurately, a function) is a value that the graph approaches.  This concept is quite different from asking "what is the value at a point on the graph" because you may have a situation where the graph is non-continuous.  Here are a couple of examples that may clear up this concept of calculus limits.

For the first example, consider the following curve.  It is a smooth, continuous curve, meaning that there are no gaps or sudden jumps in the line.  (I will do a separate post on continuity of graphs in more detail soon.)  The limit of the graph as you move along the line towards the point P (getting infinitely closer but never actually reaching), from either direction, approaches the line of y = 4.  In this case, the graph actually does have a point at y = 4, because it is continuous.  So we can technically say that the limit as f(x) approaches point P is 4, and also that f(x) = 4.

Now, let's consider a discontinuous graph to see the importance of limits.  Recall what a function actually is, and specifically in this case think about piecewise functions, and also perhaps a refresher of the vertical line test would do well here as well.  In the below discontinuous graph, we can see that it is indeed a function because it passes the vertical line test.  It also appears to be a piecewise function, in that there appears to be a different expression that defines different parts of the graph (e.g. the curve, and also the point at P).  Note that on the curve, the point P is shown with an open circle to indicate all values that approach P, but not P.  P is elsewhere on the graph.  Similarly to the continuous expression above, as the curve of the graph approaches point P and gets infinitely closer without ever reaching it, it appears to be heading towards (i.e. the limit of the graph is) 4.  However, the expression f(x) itself clearly is not the same value exactly at P, and in this case we can see that when x = p, y actually equals 10 (while the limit is 4).

I know this has been a very basic explanation of limits, but I only intend for it to serve as an introduction to this calculus concept.  I will be posting additional material on other limit-related topics, including the correct limit notation, the concepts of one-sided limits, continuity of graphs, and asymptotes, and I will present some more introductory posts that discuss tangents, velocity, and rates of change.  And of course, what is a mathematical concept without having a set of laws to go with it?  So I will present to you, for your viewing pleasure, the Limit Laws.  I hope this introduction to limits has been enough to provide you with a basic understanding of what a limit of a function is, and how it differs from a regular expression that you are used to by now.  You can't think of these the same way as you have been regarding expressions up to now.  Before, you wanted to know what was happening AT a point.  With limits, you want to know what is happening AS YOU APPROACH a point.  They sound very similar, but they are distinctly different.  A good understanding of limits is essential for you to be able to work with other calculus problems and concepts.  I hope my coming posts will be helpful for you.  As always, please remember to +1 this if it helped you.

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